1. Hierarchical Clustering
Adapted from Slides by Prabhakar Raghavan, Christopher Manning, Ray Mooney and Soumen Chakrabarti
Prasad
L14HierCluster

2. “The Curse of Dimensionality”
Why document clustering is difficult
While clustering looks intuitive in 2 dimensions, many of our applications involve 10,000 or more dimensions…
High-dimensional spaces look different
The probability of random points being close drops quickly as the dimensionality grows.
Furthermore, random pair of vectors are all almost perpendicular.
Docs contain different words.

3. Today’s Topics Hierarchical clustering Evaluation
Agglomerative clustering techniques
Evaluation
Term vs. document space clustering
Multi-lingual docs
Feature selection
Labeling

4. Hierarchical Clustering
Build a tree-based hierarchical taxonomy (dendrogram) from a set of documents.
One approach: recursive application of a partitional clustering algorithm.
animal
vertebrate
fish reptile amphib. mammal worm insect crustacean
invertebrate

5. Dendogram: Hierarchical Clustering
Clustering obtained by cutting the dendrogram at a desired level: each connected component forms a cluster.

6. Hierarchical Clustering algorithms
Agglomerative (bottom-up):
Start with each document being a single cluster.
Eventually all documents belong to the same cluster.
Divisive (top-down):
Start with all documents belong to the same cluster.
Eventually each node forms a cluster on its own.
Does not require the number of clusters k in advance
Needs a termination/readout condition
The final mode in both Agglomerative and Divisive is of no use.

7. Hierarchical Agglomerative Clustering (HAC) Algorithm
Start with all instances in their own cluster.
Until there is only one cluster:
Among the current clusters, determine the two
clusters, ci and cj, that are most similar.
Replace ci and cj with a single cluster ci  cj
Consider agglomerative clustering on n points on a line. Explain how you could avoid n3 distance computations - how many will your scheme use?

8. Dendrogram: Document Example
As clusters agglomerate, docs likely to fall into a hierarchy of “topics” or concepts. d3 d5 d1
d3,d4,d5 d4 d2
d1,d2
d4,d5 d3

9. Key notion: cluster representative
We want a notion of a representative point in a cluster, to represent the location of each cluster
Representative should be some sort of “typical” or central point in the cluster, e.g.,
point inducing smallest radii to docs in cluster
smallest squared distances, etc.
point that is the “average” of all docs in the cluster
Centroid or center of gravity
Measure intercluster distances by distances of centroids.

10. Example: n=6, k=3, closest pair of centroids
Centroid after
second step. d1 d2
Centroid after first step.

11. Outliers in centroid computation
Can ignore outliers when computing centroid.
What is an outlier?
Lots of statistical definitions, e.g.
moment of point to centroid > M  some cluster moment.
Say 10.
Centroid
Outlier

12. Closest pair of clusters
Many variants to defining closest pair of clusters
Single-link
Similarity of the most cosine-similar (single-link)
Complete-link
Similarity of the “furthest” points, the least cosine-similar
Centroid
Clusters whose centroids (centers of gravity) are the most cosine-similar
Average-link
Average cosine between pairs of elements

13. Single Link Agglomerative Clustering
Use maximum similarity of pairs:
Can result in “straggly” (long and thin) clusters due to chaining effect.
After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is:

14. Single Link Example

15. Complete Link Agglomerative Clustering
Use minimum similarity of pairs:
Makes “tighter,” spherical clusters that are typically preferable.
After merging ci and cj, the similarity of the resulting cluster to another cluster, ck, is: Ci Cj Ck

16. Complete Link Example

17. Computational Complexity
In the first iteration, all HAC methods need to compute similarity of all pairs of n individual instances which is O(n2).
In each of the subsequent n2 merging iterations, compute the distance between the most recently created cluster and all other existing clusters.
In order to maintain an overall O(n2) performance, computing similarity to each cluster must be done in constant time.
Else O(n2 log n) or O(n3) if done naively
Iteration one: n^2 distances
Computing distances from newly formed cluster = o(n)
Subsequent iterations = o(n)
So overall complexity for subsequent iterations: o(n^2)

18. Group Average Agglomerative Clustering
Use average similarity across all pairs within the merged cluster to measure the similarity of two clusters.
Compromise between single and complete link.
Two options:
Averaged across all ordered pairs in the merged cluster
Averaged over all pairs between the two original clusters
Some previous work has used one of these options; some the other. No clear difference in efficacy
Weighted average can be computed efficiently (remember avg and number of points)

19. Computing Group Average Similarity
Assume cosine similarity and normalized vectors with unit length.
Always maintain sum of vectors in each cluster.
Compute similarity of clusters in constant time:

20. Efficiency: Medoid As Cluster Representative
The centroid does not have to be a document.
Medoid: A cluster representative that is one of the documents
For example: the document closest to the centroid
One reason this is useful
Consider the representative of a large cluster (>1000 documents)
The centroid of this cluster will be a dense vector
The medoid of this cluster will be a sparse vector
Compare: mean/centroid vs. median/medoid

21. Efficiency: “Using approximations”
In standard algorithm, must find closest pair of centroids at each step
Approximation: instead, find nearly closest pair
use some data structure that makes this approximation easier to maintain
simplistic example: maintain closest pair based on distances in projection on a random line
Random line

22. Term vs. document space
So far, we clustered docs based on their similarities in term space
For some applications, e.g., topic analysis for inducing navigation structures, can “dualize”
use docs as axes
represent (some) terms as vectors
proximity based on co-occurrence of terms in docs
now clustering terms, not docs

23. Term vs. document space Cosine computation Cluster labeling
Constant for docs in term space
Grows linearly with corpus size for terms in doc space
Cluster labeling
Clusters have clean descriptions in terms of noun phrase co-occurrence
Application of term clusters
Cf. Latent Semantic Indexing

24. Multi-lingual docs E.g., Canadian government docs.
Every doc in English and equivalent French.
Must cluster by concepts rather than language
Simplest: pad docs in one language with dictionary equivalents in the other
thus each doc has a representation in both languages
Axes are terms in both languages

25. Feature selection Which terms to use as axes for vector space?
Large body of (ongoing) research
IDF is a form of feature selection
Can exaggerate noise e.g., mis-spellings
Better to use highest weight mid-frequency words – the most discriminating terms
Pseudo-linguistic heuristics, e.g.,
drop stop-words
stemming/lemmatization
use only nouns/noun phrases
Good clustering should “figure out” some of these

26. Major issue - labeling
After clustering algorithm finds clusters - how can they be useful to the end user?
Need pithy label for each cluster
In search results, say “Animal” or “Car” in the jaguar example.
In topic trees (Yahoo), need navigational cues.
Often done by hand, a posteriori.
News doc cluster labels

27. How to Label Clusters Show titles of typical documents
Titles are easy to scan
Authors create them for quick scanning!
But you can only show a few titles which may not fully represent cluster
Show words/phrases prominent in cluster
More likely to fully represent cluster
Use distinguishing words/phrases
Differential labeling
Cf. News docs appl

28. Labeling
Common heuristics - list 5-10 most frequent terms in the centroid vector.
Drop stop-words; stem.
Differential labeling by frequent terms
Within a collection “Computers”, clusters all have the word computer as frequent term.
Discriminant analysis of centroids.
Perhaps better: distinctive noun phrase

29. What is a Good Clustering?
Internal criterion: A good clustering will produce high quality clusters in which:
the intra-class (that is, intra-cluster) similarity is high
the inter-class similarity is low
The measured quality of a clustering depends on both the document representation and the similarity measure used

30. External criteria for clustering quality
Quality measured by its ability to discover some or all of the hidden patterns or latent classes in gold standard data
Assesses a clustering with respect to ground truth
Assume documents with C gold standard classes, while our clustering algorithms produce K clusters, ω1, ω2, …, ωK with ni members.

31. External Evaluation of Cluster Quality
Simple measure: purity, the ratio between the dominant class in the cluster πi and the size of cluster ωi
Others are entropy of classes in clusters (or mutual information between classes and clusters)

32. Purity example                  Cluster I Cluster II
 
 
 
 
 
 
 
  
Cluster I
Cluster II
Cluster III
Cluster I: Purity = 1/6 (max(5, 1, 0)) = 5/6
Cluster II: Purity = 1/6 (max(1, 4, 1)) = 4/6
Cluster III: Purity = 1/5 (max(2, 0, 3)) = 3/5

33. A B C D Rand Index Number of points Same Cluster in clustering
Different Clusters in clustering
Same class in ground truth A C
Different classes in ground truth B D

34. Rand index: symmetric version
Compare with standard Precision and Recall.

35. 20 24 72 Rand Index example: 0.68 Number of points
Same Cluster in clustering
Different Clusters in clustering
Same class in ground truth 20 24
Different classes in ground truth 72
20+72 / = 92 / 136 = 0.68

36. SKIP WHAT FOLLOWS

37. Evaluation of clustering
Perhaps the most substantive issue in data mining in general:
how do you measure goodness?
Most measures focus on computational efficiency
Time and space
For application of clustering to search:
Measure retrieval effectiveness

38. Approaches to evaluating
Anecdotal
User inspection
Ground “truth” comparison
Cluster retrieval
Purely quantitative measures
Probability of generating clusters found
Average distance between cluster members
Microeconomic / utility

39. Anecdotal evaluation Probably the commonest (and surely the easiest)
“I wrote this clustering algorithm and look what it found!”
No benchmarks, no comparison possible
Any clustering algorithm will pick up the easy stuff like partition by languages
Generally, unclear scientific value.

40. User inspection Induce a set of clusters or a navigation tree
Have subject matter experts evaluate the results and score them
some degree of subjectivity
Often combined with search results clustering
Not clear how reproducible across tests.
Expensive / time-consuming

41. Ground “truth” comparison
Take a union of docs from a taxonomy & cluster
Yahoo!, ODP, newspaper sections …
Compare clustering results to baseline
e.g., 80% of the clusters found map “cleanly” to taxonomy nodes
How would we measure this?
But is it the “right” answer?
There can be several equally right answers
For the docs given, the static prior taxonomy may be incomplete/wrong in places
the clustering algorithm may have gotten right things not in the static taxonomy
“Subjective”

42. Ground truth comparison
Divergent goals
Static taxonomy designed to be the “right” navigation structure
somewhat independent of corpus at hand
Clusters found have to do with vagaries of corpus
Also, docs put in a taxonomy node may not be the most representative ones for that topic
cf Yahoo!

43. Microeconomic viewpoint
Anything - including clustering - is only as good as the economic utility it provides
For clustering: net economic gain produced by an approach (vs. another approach)
Strive for a concrete optimization problem
Examples
recommendation systems
clock time for interactive search
expensive

44. Evaluation example: Cluster retrieval
Ad-hoc retrieval
Cluster docs in returned set
Identify best cluster & only retrieve docs from it
How do various clustering methods affect the quality of what’s retrieved?
Concrete measure of quality:
Precision as measured by user judgements for these queries
Done with TREC queries

45. Evaluation Compare two IR algorithms
1. send query, present ranked results
2. send query, cluster results, present clusters
Experiment was simulated (no users)
Results were clustered into 5 clusters
Clusters were ranked according to percentage relevant documents
Documents within clusters were ranked according to similarity to query

46. Sim-Ranked vs. Cluster-Ranked

47. Relevance Density of Clusters

48. Buckshot Algorithm Another way to an efficient implementation:
Cut where
You have k
clusters
Another way to an efficient implementation:
Cluster a sample, then assign the entire set
Buckshot combines HAC and K-Means clustering.
First randomly take a sample of instances of size n
Run group-average HAC on this sample, which takes only O(n) time.
Use the results of HAC as initial seeds for K-means.
Overall algorithm is O(n) and avoids problems of bad seed selection.
Uses HAC to bootstrap K-means

49. Bisecting K-means
Divisive hierarchical clustering method using K-means
For I=1 to k-1 do {
Pick a leaf cluster C to split
For J=1 to ITER do {
Use K-means to split C into two sub-clusters, C1 and C2
Choose the best of the above splits and make it permanent} }
Steinbach et al. suggest HAC is better than k-means but Bisecting K-means is better than HAC for their text experiments
Consider running 2-means clustering on a corpus, each doc of which is from one of two different languages. What are the two clusters we would expect to see?
Is agglomerative clustering likely to produce different results to the above?
Is the centroid of normalized vectors normalized?
Suppose a run of agglomerative clustering finds k=7 to have the highest value amongst all k. Have we found the highest-value clustering amongst all clusterings with k=7?